> >***** Entailment in Natural-Language Translation
> >
> >So, in natural language, here is the state of an example universe:
> >
> > Doc-1 is this string:
> >
> > The class herein called "Dog" is a
> > subclass of the class herein called
> > "Mammal".
> >
> > Claim-1a is expressed by Doc-1.
> >
> > Claim-1b: Doc-1 (as defined above) is on the web at address URL-1.
> >
> > The creator of Doc-1 and the controller of URL-1 is "Nancy" (the
> > namer).
> >
> >
> > Doc-2 is this string:
> >
> > The thing herein called "spot" is a member
> > of the class called "Dog" in the document
> > on the web at address URL-1.
> >
> > Claim-2a is expressed by Doc-2.
> >
> > Claim-2b: Doc-2 (as defined above) is on the web at address URL-2.
> >
> > The creator of Doc-2 and the controller of URL-2 is "Arthur" (the
> > author).
> >
> >
> > Proposition-1: The thing called "spot" in Doc-2 is a member
> > of the class which is called "Mammal" in the document on the web at
> > address URL-1.
> >
> > Proposition-2: The thing called "spot" in the document on the web
> > at address URL-2 is a member of the class which is called "Mammal"
> > in the document on the web at address URL-2.
> >
> >It seems clear to me that Proposition-1 logically follows from
> >Claim-1b and Claim-2a.
>
> I disagree. In fact, I will state it as a fact (not an opinion) that
> it does not follow *logically*, since neither of those claims (1b and
> 2a) use the term "Mammal", so it follows by the Craig interpolation
> lemma that proposition-1 does not follow from them.
Claim-1b does kind of use the term "Mammal", in that it includes by
reference the text of Doc-1. If we wanted be more verbose, we could
restate claim-1b as:
Claim-1b: Doc-1 is on the web at address URL-1 and is the string:
The class herein called "Dog" is a
subclass of the class herein called
"Mammal".
> So I know that the sense of 'entailment' you have in mind here is not
> logical entailment. I also do not think it is 'natural language'
> entailment, in any reasonable sense of that rather vague term, for
> much the same reasons. Lets take the example off the web and see what
> it looks like. Suppose I have Doc-1 written on a piece of paper and I
> brandish this piece of paper at someone and assert 'This piece of
> paper exists and Spot is a Dog (in the sense used on this piece of
> paper)' which is analogous to your 1b + 2a. Now, what is on the paper
> is irrelevant: it still doesn't follow *from what I said* that Spot
> is a Mammal.
Actually, it does, given my interpration of the key phrase "in the
sense used on this piece of paper". That phrase has a lot of power.
Remember my description of how one communicates at least some of the
meaning of a term? You publish a declarative statement which is true
only for certain meanings of the term. (Point "y." in the original
message.) I claim that when someone uses a phrase like "in the sense
[or meaning] used in [some declarative expression]" we know that they
agree with that expression. (subject to caveats about them negating,
quoting, or otherwise not just asserting their own statement.)
This is the heart of my argument in all this. The rest is details.
What would it mean for me to say, "Spot is a member of the class of
things called 'Dog' in Doc-1, although Doc-1 is not a true statement"
?
I'm not sure if that counts as a paradox, or just a false statement,
but it sure doesn't seem like it could be true.
I doubt you're convinced, but I can't think of how else to argue it
until I hear your objection....
<thinking...>
How could "Spot is a member of the class of things called 'Dog' in
Doc-1" be true while Doc-1 is false? "Spot is the thing called
'Rover' in Doc-3" and Doc-3 is false. I need some logic with
quotation:
Syn(Spot, "Rover", "Dog(Rover)") => X(Spot) & Syn(X, "Dog", "Dog(Rover)")
Syn(Spot, "Rover", "false") <=> false
I dunno. More work to do here.
Anyway, if we accept this, my arguement might proceed:
We're given
Doc-1 is on the web at address URL-1 and is the string:
The class herein called "Dog" is a
subclass of the class herein called
"Mammal".
and
The thing herein called "spot" is a member of the class called
"Dog" in the document on the web at address URL-1.
Assuming for the moment that there can be only one string at a given
web address, we can substitute to get:
The thing herein called "spot" is a member of the class called
"Dog" in the string, Doc-1:
The class herein called "Dog" is a
subclass of the class herein called
"Mammal".
Change the "herein"s to name Doc-1:
The thing herein called "spot" is a member of the class called
"Dog" in the string:
The class called "Dog" in Doc-1 is a
subclass of the class called
"Mammal" in Doc-1.
Take the phrase "... called X in Y ...", as discussed above, to mean
the same if X is replaced by an existentially quantifed variable and the
phrase is conjoined with Y, we get:
There exist some D and M, such that the thing
herein called "spot" is a member of D, D is a subclass of M,
D is called "Dog" in Doc-1, and M is called "Mammal" in Doc-1.
By the definition of subclass, and conjunction elimination:
There exists some M such that the thing herein called "spot" is a
member of M, and M is called "Mammal" in Doc-1.
We bring back in the naming of "here" as Doc2, and the fact that Doc-1
is on the web at URL-1 (which should have been carried along here as
more conjoined facts, but I was lazy), and do some kind of existial
elimination using the fact that "the thing called" is an inverse
function property, and we get:
> > Proposition-1: The thing called "spot" in Doc-2 is a member
> > of the class which is called "Mammal" in the document on the web at
> > address URL-1.
Q.E.D. and good night. :-)
-- sandro